# Transformations of structures: An algebraic approach

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## Abstract

This paper introduces a new mathematical approach to transformations of structures, where the concept of “structure” is extremely general. Many structures and transformations that arise in biology as well as computer science are special cases of our concepts. A structure may be changed by finding an occurrence of a pattern and replacing it by another pattern as specified by a rule. To prove theorems about long sequences of applications of complicated rules, we need precise and tractable mathematical definitions of rules and how to apply them. This paper presents such definitions and three fundamental theorems, together with brief remarks on applications to control flow analysis, record handling, and evaluation of recursively defined functions. Unlike some previous efforts toward a rigorous theory of transformations of structures, this paper uses ideas and results from abstract algebra to minimize the need for elaborate constructions.

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### References

- 1.M. A. Arbib and E. G. Manes,
*Arrows, Structures, and Functors*, Academic Press, New York, 1975.Google Scholar - 2.G. Berry and J.-J. Levy, Minimal and optimal computations of recursive programs,
*J. ACM*26 (1979), 148–175.Google Scholar - 3.G. Birkhoff and J. D. Lipson, Heterogeneous algebras,
*J. Combinatorial Theory*, 8 (1970), pp. 115–133.Google Scholar - 4.V. Claus, H. Ehrig, and G. Rozenberg (Editors),
*Graph Grammars and their Application to Computer Science and Biology*, Lecture Notes in Computer Sci., 73 (1979).Google Scholar - 5.E. F. Codd,
*A relational model of data for large shared data banks*, Comm. ACM, 13 (1970), pp. 377–387.Google Scholar - 6.J. A. Darringer and W. H. Joyner,
*A new look at logic synthesis*, IBM Research Report RC 8268, Yorktown Heights NY, May 1980.Google Scholar - 7.H. Ehrig and H.-J. Kreowski, Contributions to the algebraic theory of graph grammars,
*Math. Nachr.*, to appear.Google Scholar - 8.H. Ehrig and H.-J. Kreowski, Applications of graph grammar theory to consistency, synchronization, and scheduling in data base systems,
*Information Sciences*, to appear.Google Scholar - 9.H. Ehrig, H.-J. Kreowski, A. Maggiolo-Schettini, B. K. Rosen, and J. Winkowski, Deriving structures from structures,
*Lecture Notes in Computer Science*, 64 (1978), pp. 177–190.Google Scholar - 10.H. Ehrig, M. Pfender, and H. J. Schneider, Graph grammars: an algebraic approach,
*Proc. 14th Ann. IEEE Symp. on Switching and Automata Theory*, Iowa City, October 1973, pp. 167–180.Google Scholar - 11.H. Ehrig and B. K. Rosen,
*Commutativity of independent transformations on complex objects*, IBM Research Report RC 6251, Yorktown Heights NY, October 1976.Google Scholar - 12.H. Ehrig and B. K. Rosen, The mathematics of record handling,
*SIAM J. Computing*, 9 (1980), 441–469.Google Scholar - 13.H. Ehrig and B. K. Rosen,
*Commutativity, parallelism, and concurrency for transformations of structures*, Bericht 79–21, Fachbereich Informatik, Tech. U. Berlin, October 1979.Google Scholar - 14.H. Ehrig and K. W. Tischer, Graph grammars and applications to specialization and evolution in biology,
*J. Computer and System Sci.*, 11 (1975), pp. 212–236.Google Scholar - 15.S. Eilenberg and J. Wright, Automata in general algebras,
*Information and Control*, 11 (1967), 452–470.Google Scholar - 16.R. Farrow, K. Kennedy, and L. Zucconi, Graph grammars and global program data flow analysis,
*Proc. 17th Ann. IEEE Symp. on Foundations of Computer Sci.*, Houston, October 1976, pp. 42–56.Google Scholar - 17.H. Herrlich and G. Strecker,
*Category Theory*, Allyn and Bacon, Rockleigh, New Jersey, 1973.Google Scholar - 18.H.-J. Kreowski,
*Ein Pumpinglemma für Kanten-Kontextfreie Graph-Sprachen*, Bericht 77–15, Fachbereich Informatik, Tech. U. Berlin, September 1977. (See also [4].)Google Scholar - 19.M. J. O'Donnell, Computing in systems described by equations,
*Lecture Notes in Computer Science*, 58 (1977).Google Scholar - 20.G. Pacini, C. Montangero, and F. Turini, Graph representation and computation rule for typeless recursive languages,
*Lecture Notes in Computer Science*, 14 (1974), pp. 157–169.Google Scholar - 21.Padawitz, P.,
*Graph grammars and operational semantics*, Bericht 78–33, Fachbereich Informatik, Tech. U. Berlin, October 1978. (See also [4].)Google Scholar - 22.V. Rajlich, Dynamics of discrete systems ⋯,
*J. Computer and System Sci.*, 11 (1975), pp. 186–202.Google Scholar - 23.B. K. Rosen, Deriving graphs from graphs by applying a production,
*Acta Informatica*, 4 (1975), pp. 337–357.Google Scholar - 24.H. J. Schneider and H. Ehrig, Grammars on partial graphs,
*Acta Informatica*, 6 (1976), pp. 297–316.Google Scholar - 25.J. Staples, A class of replacement systems with simple optimality theory,
*Bull. Austral. Math. Soc.*, 17 (1977), p. 335–350.Google Scholar - 26.J. Staples, Computation on graph-like expressions,
*Theoretical Computer Sci.*, 10 (1980), 171–185.Google Scholar - 27.J. Staples, Optimal evaluations of graph-like expressions,
*Theoretical Computer Sci.*, 10 (1980), 297–316.Google Scholar - 28.J. Staples, Speeding up subtree replacement systems,
*Theoretical Computer Sci.*, 11 (1980), 39–47.Google Scholar - 29.S. A. Vere, Relational production systems,
*Artificial Intelligence*, 8 (1977), pp. 47–68.Google Scholar - 30.J. Vuillemin, Correct and optimal implementations of recursion in a simple programming language,
*J. Computer and System Sci.*, 9 (1974), pp. 332–354.Google Scholar